The purpose of § is to upgrade some classical constructions from the theory of exponential sums on tori to more general varieties.
In §REF -REF  we recall the basic notions of Artin-Schreier-Witt theory, and give an alternative formulation in terms of the flat liftings used in [1]}. Our reason for working with flat liftings is that they lead to a very general theory of Dwork splitting functions. Such splitting functions are an integral part of the classical theory of exponential sums (see [2]} and [3]}), and can be constructed quite explicitly from the Artin-Hasse exponential series (as in [4]}). However,
these classical splitting functions are inapplicable to the study of exponential sums on more general varieties. In §REF  we construct
splitting functions using a generalized Artin-Hasse exponential. This generalized
Artin-Hasse exponential is “compatible” with a given lifting of Frobenius (as opposed to
the classical Artin-Hasse exponential, which is compatible with the Frobenius \(t \rightarrow t^p\) ). In particular, it allows us to explicitly define splitting functions over any
variety.
This approach to splitting functions appears to be genuinely new, and allows us to deduce, for example, that the \(F\) -crystal associated to any character of \(\pi _1(X)\)  (not necessarily finite) factoring through a \(\mathbb {Z}_p\) -tower is free. This result is essential in order to apply the main theorems of [1]}.
